Problem: A circle has a sector with area $\dfrac{189}{8}\pi$ and central angle $105^\circ$. What is the area of the circle? ${81\pi}$ $\color{#9D38BD}{105^\circ}$ ${\dfrac{189}{8}\pi}$
Answer: The ratio between the sector's central angle $\theta$ and $360^\circ$ is equal to the ratio between the sector's area, $A_s$ , and the whole circle's area, $A_c$ $\dfrac{\theta}{360^\circ} = \dfrac{A_s}{A_c}$ $\dfrac{105^\circ}{360^\circ} = \dfrac{189}{8}\pi \div A_c$ $\dfrac{7}{24} = \dfrac{189}{8}\pi \div A_c$ $A_c \times \dfrac{7}{24} = \dfrac{189}{8}\pi$ $A_c = \dfrac{189}{8}\pi \times \dfrac{24}{7}$ $A_c = 81\pi$